INTEGRATION OF DIFFERENTIAL EQUATIONS. 117 



If we make k K m , this will become 



which may be integrated any number of times for K, and con- 

 sequently, if it is multiplied by any rational and integral func- 



tion of K, it may still be integrated by parts as often as we 



flfc 

 please. Now -7 will become mK^ s ~ l dK] and as m and s are 



& 



integral, the method of parts applies, provided s is not greater 

 than m 1, which it is in none of our formulas. 



Fourier's expression, by means of definite integrals for the 

 t th differential coefficient of any function, would enable us to 

 extend our solutions to the cases in which p is fractional. But 

 merely analytical transformations of the results at which we 

 have arrived are not of much interest, and the methods of effect- 

 ing them are direct and obvious. 



Equation (1) admits of another symbolical solution besides 

 the one already given. 



It is easily seen, that if 



SX = *"* JL _L . Jl _L. . . . Itjf, (p factors), 



a n = (n pm -f 1) . . . (n m + 1) 5 n , 



which is what (3) is, when f(k) = 1. (-r applies to all that 

 follows it.J 



Hence we shall clearly have 



^ d 1 d I * d 1 Y 



y ~ dx^dxtf^- "Txx-^ 

 for the solution of (1). 



Similarly, the solution of (6) is 



,, _ .<*-i)-H-i) _ 1 T 



y ~ 'dxx- 1 " "dxx 1 



Many applications and modifications of the method we have 

 employed will readily present themselves, but the subject is not 

 of sufficient importance to deserve a fuller discussion. It is not 

 difficult to multiply artifices, by means of which particular 

 equations may be solved, but the results will, generally speak- 

 ing, be of little value. 



